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Graduate Summer School 2017

Random Matrices

The Graduate Summer School bridges the gap between a general graduate education in mathematics and the specific preparation necessary to do research on problems of current interest. In general, these students will have completed their first year of graduate school, and in some cases, may already be working on a thesis. While a majority of the participants will be graduate students, some postdoctoral scholars and researchers may also be interested in attending.

The main activity of the Graduate Summer School will be a set of intensive short courses offered by leaders in the field, designed to introduce students to exciting, current research in mathematics. These short courses will not duplicate standard courses available elsewhere. Each course will consist of lectures with problem sessions. Course assistants will be available for each lecture series. The participants of the Graduate Summer School meet three times each day for lectures, with one or two problem sessions scheduled each day as well.

Random matrix theory has become a vital tool in many areas of mathematics and related fields like physics, computer science, statistics, and data science. Recent years have seen the development of a battery of beautiful mathematical results and applications in this area. The school will cover a range of topics including universality of spectral statistics, high dimensional geometry and concentration methods, free probability, the Kardar-Parisi-Zhang universality class, orthogonal polynomials, Riemann-Hilbert Problems and Painleve transcendents, applications of random matrices in statistical physics, topological recursions.

Student preparation: We seek motivated students interested in the mathematical aspects of random matrix theory and related fields. This includes mathematically-inclined students from physics, computer science, statistics, applied mathematics, and related areas. Though familiarity with some of the topics listed above would be helpful, the formal prerequisites are limited to the content of standard introductory courses in linear algebra, and probability (plus some familiarity with ODEs and PDEs will help).

The 27th Annual PCMI Summer Session will be held June 25 – July 15, 2017.

Eigenvalues, graphs, and Catalan numbers: the semicircle law and beyond

The most basic, and hence arguably most famous result in random matrix theory is the (Wigner) semicircle law. Simply stated, for any Wigner symmetric/Hermitian random matrix with independent, identically distributed entries of mean 0 and variance 1, the distribution of a randomly-chosen eigenvalue converges to the (normalized) semicircle. This result is true regardless of the entry distribution (with only very minor conditions). The proof involves linear algebra and combinatorics: in particular, looking at traces of powers of the matrix and counting walks on the complete graph to obtain Catalan numbers. We will prove this result and then delve into why much more can actually be shown. From the fact that the shape of the histogram for all eigenvalues is semicircular, to investigating the fluctuations from the semicircle, we will explore the surprisingly stable structures that emerge from the spectrum of large random matrices.

How many equilibria will a large complex system, modeled by N randomly coupled autonomous nonlinear differential equations typically have? How many of those equilibria are stable, that is are local attractors of the nearby trajectories? These questions arise in many applications and can be partly answered by employing the methods of Random Matrix Theory. The lectures will outline these recent developments.

As observed by Dyson and Wigner, instances of classical random matrix ensembles (such as the Gaussian Unitary Ensemble, Gaussian Orthogonal Coulomb Gases, Ginibre Ensemble) can also be viewed as systems of particles in the plane or on the real line with logarithmic or Coulomb interactions, at particular temperatures, which are also called beta-ensembles. These have been in recent years intensely studied for their own sake. We will examine general Coulomb and Log gases (including in higher dimension than 2), taking a point of view based on the detailed expansion of the interaction energy. This allows us to describe the macroscopic and microscopic behavior of the systems. In particular we will show a Large Deviations Principle for the empirical field and a Central Limit Theorem for fluctuations down to the mesoscopic scales. This allows us to observe the effect of the temperature as it gets very large or very small, and to connect with crystallization questions, such as the occurence of the triangular Abrikosov lattice. The main results are joint with Thomas Leble and also based on previous works with Etienne Sandier, Nicolas Rougerie and Mircea Petrache.

Voiculescu invented his free probability theory to approach problems in von Neumann algebras. A key feature of his theory is the treatment of free independence — based on the notion of free products, such as free products of groups — as a surprisingly close parallel to classical independence. Rather unexpectedly it turned out that there are deep connections between his theory and the theory of random matrices: very roughly, free probability describes certain aspects of asymptotic behavior of random matrix models. In this course, we will start with an introduction to free probability theory, discuss connections with random matrix theory, and finally describe some applications of results from random matrices in operator algebras and vice versa.

In recent years, Wigner matrices have been studied in increasing generality by gradually relaxing the original conditions that required independent, identically distributed entries. We analyze the key equation, the so-called matrix Dyson equation, that governs the density of states and the behavior of resolvent matrix elements of the corresponding ensemble. As an application, we present local laws and local spectral universality for random matrices with correlated entries.

The lecture series will present some of the advances in the last few years (by Erdos, Schlein, Yau, Yin, Vu, myself, and others) in demonstrating universal limiting asymptotics for the spectrum of random matrices, focusing particularly on the Lindeberg exchange strategy and on the circular law.

The Kardar-Parisi-Zhang equation is a canonical model for one dimensional random growth. It is also a member of a large universality class containing also directed polymer free energies, and randomly forced fluids in one dimension, characterized by nonstandard fluctuations, many of which were first discovered in random matrix theory. We will introduce the main ideas behind the meaning of the equation and obtaining exact asymptotic distributions.

Consider a random matrix with i.i.d. normal entries. Since its distribution is invariant under rotations, any normalized eigenvector is uniformly distributed over the unit sphere. For a general distribution of the entries, this is no longer true. Yet, if the size of the matrix is large, the eigenvectors are distributed approximately uniformly. This property, called delocalization, can quantified in various senses. In these lectures, we will discuss recent results on delocalization for general random matrices.

The behavior of random matrices can be understood by studying their limits as the dimension tends to infinity. The limiting operators
can be used to study eigenvalue statistics. This mini-course will be an elementary introduction to this theory.

Among others, we will give a simple description of the Tracy-Widom distribution and derive some basic properties. We will also give a non-computational proof of the Wigner semicircle law. We will discuss several open problems. No previous experience with operators is required.